Planning could (and have in the past) ask for a CoL to be submitted, so yes they (the planners) can do something about it.
Can they?
Where in the TCPA, or various GPDO's and amendments, or more specifically the GDPO 1995 does it state that this can be done?
You may have been asked/requested, but I don't know of any authority for the planners to actually require a LDC from anyone.
In fact, it is up to the homeowner (or anyone) to make a LDC application (for their own benefit) and not for the planners to require one (in order for the planners to determine lawful development for their own purpose or because of a complaint).
If it is deemed that unauthorised/unlawful development has taken place then planners can either take enforcement action or give the opportunity for a retrospective planning application to be made, not a LDC application
The whole idea of a LDC is to give assurance to the homeowner that the development is lawful, or for a third party to have information as to whether development is unlawful. It is not for the purpose of planners "checking" the development as a result of a complaint.
So in the context of this thread, if the OP's neighbour complains then if the planner decides to investigate there will be only two outcomes
1. The development is found to be PD and no further action can (by law) be taken
or
2.The development is found not to be PD and a full planning application will be required - or enforcement action taken
By all means cite the relevant planning law to contradict this, but that's the situation AFAIK